Precook the broccoli - Bring a large pot of water to a boil and salt it generously. Add the florets to the boiling water and cook until very tender, about 10 minutes. Once tender, remove the cooked broccoli with a slotted spoon and set aside. Reserve the broccoli "broth."
Begin the sauce - While the broccoli is cooking, pour ½ cup olive oil into a large pan over medium-low heat. Peel and slice the garlic, add it to the pan, and sauté it until it turns golden (about 2-3 minutes), Add 1 t red chili flakes and cook for another 30 seconds. Add the broccoli to the pan.
Cook the pasta - Bring the broccoli "broth " to a boil, add the pasta, and cook until it is just al dente, a minute or so shy of the package directions.
Finish the sauce - Add 1 cup of the pasta water to the pan to make the broccoli sauce and continue cooking at a low simmer. For a creamier sauce, mash the broccoli against the side of the pan.
Combine - Add the pasta to the pan and cook, stirring frequently, until the pasta is perfectly al dente. Stir in the beans, if using, to warm them through.
Finish - Remove the pan from the heat and add the pecorino. Toss or mix well to coat. If needed, add more reserved pasta water to achieve a loose consistency. Serve with more pecorino, red chili flakes, and a drizzle of your best extra virgin olive oil. Enjoy!